Question

Find the roots of the quadratic equation \( 2x^2 - 4x - 6 = 0 \).

• A. \( x = 1 \, \text{or} \, x = -3 \)
• B. \( x = -1 \, \text{or} \, x = 3 \)
• C. \( x = 2 \, \text{or} \, x = -1 \)
• D. \( x = 3 \, \text{or} \, x = -2 \)

Hint:

Remember: For a quadratic equation \( ax^2 + bx + c = 0 \), use the quadratic formula to find the roots. The discriminant \( b^2 - 4ac \) determines the nature of the roots.

Answer 1

The Correct Option is A

Step 1: Apply the quadratic formula The standard quadratic formula for \( ax^2 + bx + c = 0 \) is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For \( 2x^2 - 4x - 6 = 0 \), the coefficients are:- \( a = 2 \),- \( b = -4 \),- \( c = -6 \).Step 2: Substitute coefficients into the formula Plugging in \( a = 2 \), \( b = -4 \), and \( c = -6 \) yields:\[x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 2 \times (-6)}}{2 \times 2}\]\[x = \frac{4 \pm \sqrt{16 + 48}}{4}\]\[x = \frac{4 \pm \sqrt{64}}{4}\]\[x = \frac{4 \pm 8}{4}\]Step 3: Calculate the roots The two solutions for \( x \) are:\[x = \frac{4 + 8}{4} = \frac{12}{4} = 3\]and\[x = \frac{4 - 8}{4} = \frac{-4}{4} = -1\]Answer: The roots of the equation are \( x = 3 \) or \( x = -1 \). This corresponds to option (1).